Chapter 1.7, Problem 103E

Photography A photographer takes a picture of a three-foot-tall painting hanging in an art gallery. The camera lens is 1 foot below the lower edge of the painting (see figure). The angle $\beta$ subtended by the camera lens $x$ feet from the painting is given by

$\beta =\arctan \frac{3x}{x^2+4},x>0.$

(a) Use a graphing utility to graph $\beta$ as a function of $x$.

(b) Use the graph to approximate the distance from the picture when $\beta$ is maximum.

(c) Identify the asymptote of the graph and interpret its meaning in the context of the problem.